Weaver • The Imperfections of Science 



Logic is indeed an integral and cen- 

 tral part of science. But logic, although 

 a vastly useful mental tool, does not 

 now have the reputation which it was 

 once supposed to deserve. 



There are two main types of logic: 

 deductive and inductive. In the former, 

 one starts by making a certain num- 

 ber of pure assumptions— technically 

 speaking, he adopts the postulates of 

 the system under examination. Then 

 with the addition of a certain accepted 

 vocabular)' of signs, certain assumed 

 formation rules for combining the 

 signs, and certain assumed transforma- 

 tion rules for deriving new formulas 

 from old ones— with this assumed ma- 

 chinery one then proceeds to— to do 

 what? 



Of course, all he can possibly do is 

 to unroll, in all its lovely and unsus- 

 pected complexity, the truths— or more 

 properly, the formally correct relation- 

 ships—which were inherent in what he 

 originally assumed. This procedure is, 

 of course, quite powerless to create 

 truths— it can only reveal what has 

 been previously and unconsciously as- 

 sumed. 



But apart from this inherent limita- 

 tion on deductive logic, which has of 

 course been long recognized, there 

 have rather recently been discovered, 

 by Godel, wholly unsuspected and 

 startling imperfections in any system 

 of deductive logic. Godel has obtained 

 two main results. He proved that it is 

 impossible— theoretically impossible, 

 not just unreasonably difficult- to 

 prove the consistency of any set of 

 postulates which is so to speak, rich 

 enough in content to be interesting. 

 The question, "Is there an inner flaw 

 in this system?" is a question which is 

 simply unanswerable. 



He also proved that any such de- 

 ductive logical system inevitably has a 

 further great limitation. Such a system 

 is essentially incomplete. Within the 



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system it is always possible to ask ques- 

 tions which are undecidable. 



If deductive logic has these vital 

 and built-in hmitations, how about in- 

 ductive logic, the branch of reasoning 

 which examines all the observed cases 

 recorded in the evidence, and seeks to 

 induce therefore general laws. To quote 

 from my previous paper on this sub- 

 ject: 



Over 200 years ago Da\'id Hume 

 bluntly denied the propriety of induc- 

 tive logic. Ever since, certain skeptics 

 have urged the necessity of practicing in- 

 duction without pretending that it has 

 any rational foundation; certain deduc- 

 tionists have vainly tried to prove Hume 

 wrong; certain philosophers have opti- 

 mistically hoped that a mild and friendly 

 attitude towards such words as "rational" 

 and "reasonable" could of itself sanction 

 their application to statements referring 

 to future and hence unexamined cases; 

 and certain scientists have felt that it is 

 vaguely sensible to suppose that future 

 phenomena would conform to past regu- 

 larities. 



Deep and troublesome questions are 

 involved here. Consider, just for a mo- 

 ment, the question: When and why does 

 a single piece of past evidence give use- 

 ful information about a future situation? 

 If one takes a single piece of copper and 

 determines that it conducts electricit\', 

 then it seems sensible to suppose that 

 other future pieces of copper will also 

 conduct electricity. But if we pick out 

 a man at random and determine that his 

 name is John, this does not at all lend 

 credence to the idea that all other men 

 are named John. The first of these seems 

 to lead to a "lawlike statement," and the 

 second to an "unlawlike" one; but no one, 

 so far as I know, has ever been able to give 

 workable form to this distinction. 



In fact, in spite of many attempts to 

 make induction intellectually tolerable, 

 the matter remains a mess. 



As the fifth imperfection in science 

 I come to a topic which, because of its 

 depth and subtlety, deserves a far more 



